i2.5. ELIMINATING L 5 (2) ON THE 10-DIMENSIONAL MODULE 819LT ~ (Gi, G3) ~ Na(X), so as M = !M(LT) and 02 (X) =f. 1, Gi ~ M, contrary
to 12.4.2.2. This completes the proof that Ki = Li if i = 3 or 4.
Finally we treat the case i = 6. Then either K 6 = L 6 as required, or as
L5/02(L5) ~ L4(2), we obtain K5/02(K5) ~ L5(2), M24, or J4 from A.3.12. The
latter three cases are impossible, since L5 acts as nt(2) on Vi, and this action
does not extend to any 6-dimensional module for L5(2), while M 24 and J4 have no
nontrivial modules of dimension 6. This completes the proof of 12.5.3. DLEMMA 12.5.4. G3 ~ M 2: G5.
PROOF. Let i := 3 or 6. By 12.5.3, Li ::::] Gi, so as NaL(v;)(AutLJ\li)) =
AutMi (Iii), Gi = MiCa(\li). Thus to show Gi ~ M, it suffices to show Ca(lli) ~ M.
If Ca(lli) acts on Li, it acts on (Lfi) = L, so that Ca(lli) ~ Na(L) = M. Thus
we may assume Ca(lli) i. Na(Li).Set Gi := Gi/02(Gi). By 12.5.3, Out(Ki) is a 2-group, so Gi = KiTOa 1 (Ki).
Thus as Ca(lli) ~ Gi, and T and Ca 1 (Ki)) act on 02 (Li0 2 (Ki)) =Li, we may
assume the preimage Y in Ki of the projection of Gin Ki with respect to the
decomposition Ki x Car (Ki) is not contained in Mi. Therefore Ki =/:-Li. As
T ~ Gi, [TnKi, Y] ~[Gin Ki, Y] ~ Yn Gi.Suppose that case (3) of 12.5.3 holds, and set Yi := 031 (NL 1 (Iii)). Then YiT
is a minimal parabolic of LiT, so as Li i. Gi, YiT = Mi n LiT = Pi. Then as
[GinKi, Y] ~ YnGi and Yi acts irreducibly as 8L2(3) on S1(Ki)* ~ E4g, Y ~ Gi
and Y = Pin Ki or S1(Ki)(Pi n Ki); as Y i. Mi, the latter case holds. ThenS7(Ki) ~ (Yl) ~ Li as Y normalizes Li by 12.5.3, contrary to m1(Li) = 1.
Therefore Ki ~ L 5 (2), M2 4 , or J4. Recall from the discussion before 12.5.3
that Mi n Mi = P Pi is the product of the two minimal parabolics P and Pi, and
02 (P) ~Ki. Then Y is a proper overgroup of 02 (P)0^2 (Pt)(TnK) which does
not contain Li. Let Yo:= (0^2 (P)Y) and recall that the discussion during the proof
of 12.5.3 determined the embedding of Mi in Ki. If Ki ~ M24, the conditions
above on Y* force Y* /0 2 (Y*) ~ S 6 and Y = Y 0 • If K* ~ L5(2), then Y/02(Y) ~
L4(2) or L3(2) x L2(2), and Yo/02(Yo) ~ L4(2) or L3(2), respectively. If K* ~ J4,then Y/02(Y) ~ M22 or 83 x 85, and Yo/02(Yo) ~ M22 or 85, respectively. In
particular, in each case Yo i. M, since Mi = PLi. Further as [Y, Gin Y] ~ Gi,
Yo ~ Gi. But as 02 (P) ~ Li ::::] Gi by 12.5.3, Yo ~ Li ~ M, contrary to the
previous remark. This contradiction completes the proof of 12.5.4. D
LEMMA 12.5.5. (1) L has two classes of involutions with representatives ji and
j 2 , where m([r,ji]) = i, m([V,j 2 ]) = 4, and Cv(ji) = V4 + V5 is of codimension 3in V.
. (2) L has two orbits on the points of V with representatives Vi and
V{ := (1 /\ 2 + 3 /\ 4).
(3) CL(V{) is R5 extended by 85.
PROOF. These are straightforward calculations.In the remainder of the section, we let V{ be defined as in 12.5.5.2
LEMMA 12.5.6. r(G, V) > 3 = m(Mv, V) = s(G, V).
D
PROOF. By 12.5.5.1, m(Mv, V) = 3, so r(G, V) 2: 3 by Theorem E.6.3. Fur-
ther if U ~ V with m(V/U) = 3 and Ca(U) i. M, then U is conjugate to Cv(ji)